James W. Brennan

Up until now, we have just been talking about
manipulating algebraic expressions. Now it is time to talk about equations. An expression is just a
statement like

2x + 3

This expression might be equal to any number, depending on the choice of x.
For example, if x = 3 then the value of this expression is 9.
But if we are writing an equation, then we are making a statement about its
value. We might say

2x + 3 = 7

A mathematical equation is either true or false. This equation, 2x + 3 = 7,
might be true or it might be false; it depends on the value chosen for x.
We call such equations conditional, because their truth depends on
choosing the correct value for x. If I choose x = 3,
then the equation is clearly false because 2(3) + 3 = 9,
not 7. In fact, it is only true if I choose x = 2. Any other
value for x produces a false equation. We say that x = 2
is the solution of this equation.

The solution of an
equation is the value(s) of the variable(s) that make the equation a true
statement.

An equation like 2x + 3 = 7 is a simple type
called a linear equation in one variable. These will always have one solution,
no solutions, or an infinite number of solutions. There are other types of
equations, however, that can have several solutions. For example, the equation

This is the normal case, as in our example where the equation 2x + 3 = 7
had exactly one solution, namely x = 2. The other two cases,
no solution and an infinite number of solutions, are the oddball cases that you
don’t expect to run into very often. Nevertheless, it is important to know that
they can happen in case you do encounter one of these situations.

This equation is obviously
true for every possible value of x. This is, of course, a ridiculously
simple example, but it makes the point. Equations that have this property are
called identities. Some examples of identities would be

2x = x + x

3 = 3

(x – 2)(x + 2) = x2
– 4

All of these equations are true for any value
of x. The second example, 3 = 3, is interesting because it
does not even contain an x, so obviously its truthfulness cannot depend
on the value of x! When you are attempting to solve an equation
algebraically and you end up with an obvious identity (like 3 = 3),
then you know that the original equation must also be an identity, and
therefore it has an infinite number of solutions.

There is no possible value
for x that could make this true. If you take a number and add 4 to it,
it will never be the same as if you take the same number and add 3 to it. Such
an equation is called a contradiction, because it cannot ever be true.

If you are attempting to solve such an equation, you will
end up with an extremely obvious contradiction such as 1 = 2. This
indicates that the original equation is a contradiction, and has no solution.